Last Minute

So we had assignment 2 due yesterday, which I handed in on the last minute. I checked the time, it was 11.58pm. Woo! No pressure. No pressure at all.

So in a2, i understood question 1 after the TA explained how the question should be answered in 3 parts. And i am glad that we didn't have to show proof of how the fibonaaci sequence ended up with the golden ratio which was what i was really confused about. But in question 2, I didn't quite understand how to prove that the function is in Theta of lg(n). What i did was prove that two values, one larger than n, and one lower than n that are powers of 2 to be elements of the Order of lg(n) and Omega of lg(n) respectively using the closed form for T(n). Due to that reason and that fact that T(n) is a non-decreasing function, i reasoned that T(n) is an element of Theta of lg(n).
I don't know to what degree this proof is correct, but it kind of makes sense to me.

Something that I did not understand however was the significance of the Max and the floors and ceilings. Since T(n) is an increasing function, wouldn't the Max always be the ceiling of (n/2)? I did not quite know of how to address that in the assignment. So i'm really looking forward to seeing the answers to this assignment.